Maxima and Minima - Examples

Examples

• The function x2 has a unique global minimum at x = 0.
• The function x3 has no global minima or maxima. Although the first derivative (3x2) is 0 at x = 0, this is an inflection point.
• The function has a unique global maximum at x = e. (See figure at right)
• The function x-x has a unique global maximum over the positive real numbers at x = 1/e.
• The function x3/3 − x has first derivative x2 − 1 and second derivative 2x. Setting the first derivative to 0 and solving for x gives stationary points at −1 and +1. From the sign of the second derivative we can see that −1 is a local maximum and +1 is a local minimum. Note that this function has no global maximum or minimum.
• The function |x| has a global minimum at x = 0 that cannot be found by taking derivatives, because the derivative does not exist at x = 0.
• The function cos(x) has infinitely many global maxima at 0, ±2π, ±4π, …, and infinitely many global minima at ±π, ±3π, ….
• The function 2 cos(x) − x has infinitely many local maxima and minima, but no global maximum or minimum.
• The function cos(3πx)/x with 0.1 ≤ x ≤ 1.1 has a global maximum at x = 0.1 (a boundary), a global minimum near x = 0.3, a local maximum near x = 0.6, and a local minimum near x = 1.0. (See figure at top of page.)
• The function x3 + 3x2 − 2x + 1 defined over the closed interval (segment) has two extrema: one local maximum at x = −1−√15⁄₃, one local minimum at x = −1+√15⁄₃, a global maximum at x = 2 and a global minimum at x = −4.